A208436 -id:A208436 - cp's OEIS frontend

A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function.

14, 21, 25, 30, 33, 38, 41, 43, 48, 50, 53, 56, 59, 61, 65, 67, 70, 72, 76, 77, 79, 83, 85, 87, 89, 92, 95, 96, 99, 101, 104, 105, 107, 111, 112, 114, 116, 119, 121, 123, 124, 128, 130, 131, 133, 135, 138, 140, 141, 143, 146, 147, 150, 151, 153, 156, 158, 159, 161

Offset: 1

N. J. A. Sloane

"All these zeros of the form s + it have real part s = 1/2 and are simple. Thus the Riemann hypothesis is true at least for t < 3330657430697." - Wedeniwski
From Daniel Forgues, Jul 24 2009: (Start)
All nontrivial zeros on the critical line, of the form 1/2 + i*t, have an associated conjugate nontrivial zero of the form 1/2 - i*t.
Any nontrivial zeros off the critical line, if ever found, would come in pairs (1/2 +- delta) + i*t, 0 < delta < 1/2. Each of these pairs, again if ever found, would then have their associated conjugate pair (1/2 +- delta) - i*t, 0 < delta < 1/2. (End)
The sequence is not strictly increasing. - Joerg Arndt, Jan 17 2015
The fraction of numbers n such that a(n) = a(n-1) has density 1. There are only finitely many numbers n with a(n) > a(n-1) + 1, see A208436. - Charles R Greathouse IV, Mar 07 2018
Conjecture: Noninteger rationals of the form m/2^bigomega(m) that can be used to approximate this sequence, i.e. a(n) ~~ 2*Pi*A374074(n)/2^bigomega(A374074(n)) - n/2 +- (...), where '~~' means 'close to'. - Friedjof Tellkamp, Jul 04 2024

			The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453).

Gregory Benford, Gravity's whispers, Futures Column, Nature, 446 (Jul 15 2010), p. 406. [Gravity waves are detected on Earth that turn out to contain a list of the zeros of the Riemann zeta function, essentially this sequence]
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E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press NY 1986.

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T. H. Chan, Pair Correlation of the zeros of the Riemann zeta function in longer ranges, arXiv:math/0305340 [math.NT], 2003.
H. T. Chan, Distribution of the zeros of the Riemann zeta function in longer intervals, arXiv:math/0305341 [math.NT], 2003.
H. T. Chan, More precise Pair Correlation Conjecture, arXiv:math/0206293 [math.NT], 2002.
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A. Y. Cheer & D. A. Goldston, Simple Zeros of the Riemann Zeta-Function, Proc. Am. Math. Soc. 118 (1993) 365.
Y.-J. Choie et al., On Robin's criterion for the Riemann Hypothesis, arXiv:math/0604314 [math.NT], 2006.
Y.-J. Choie et al., On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol 19, No. 2 (2007), 357-372.
B. Cipra, A Prime Case of Chaos
J. B. Conrey, The Riemann Hypothesis, Notices of the AMS, Vol. 50, No. 3 (2003), 341-353.
J. B. Conrey & G. Myerson, On the Balazard-Saias criterion for the Riemann Hypothesis, arXiv:math/0002254 [math.NT], 2000.
E. S. Croot, Notes on the Riemann Zeta Function (Functional Equation)
C. Daney, Open Questions:The Riemann Hypothesis
H. Delille, L'Hypothèse de Riemann (Expository papers in French) [popup windows]
J. Derbyshire, Prime Obsession
E. Elizalde, V. Moretti & S. Zerbini, On recent strategies proposed for proving Riemann hypothesis, arXiv:math-ph/0109006, 2001.
E. Elizalde, V. Moretti, and S. Zerbini, On Recent Strategies Proposed for Proving the Riemann Hypothesis, International Journal of Modern Physics A, Vol. 18, No. 12 (2003), 2189-2196.
D. W. Farmer, Counting distinct zeros of the Riemann zeta-function, The Electronic Journal of Combinatorics, 2 (1995), #R1.
K. Ford & A Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, arXiv:math/0405459 [math.NT], 2004.
K. Ford and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, Journal für die reine und angewandte Mathematik, Vol. 2005, No. 579 (2005), 145-158.
W. F. Galway, Computations related to the Riemann Hypothesis
R. Garunkstis and J. Steuding, On the distribution of zeros of the Hurwitz zeta-function, Math. Comput. 76 (2007), 323-337.
R. Garunkstis and J. Steuding, Questions around the Nontrivial Zeros of the Riemann Zeta-Function. Computations and Classifications, Math. Model. Anal. 16 (2011), 72-81.
D. A. Goldston, Notes on Pair Correlation of Zeros and Prime Numbers, arXiv:math/0412313 [math.NT], 2004.
D. A. Goldston & S. M. Gonek, A note on S(T) and the zeros of the Riemann zeta-function, arXiv:math/0511092 [math.NT], 2005.
S. Gonek, Three Lectures on the Riemann Zeta-Function, arXiv:math/0401126 [math.NT], 2004.
J. Good & B. Churchhouse, A New Conjecture Related to the Riemann Hypothesis
X. Gourdon, The 10^13 first zeros of the Riemann Zeta function and zeros computation at very large height
X. Gourdon & P. Sebah, The Riemann Zeta-function zeta(s)
S. W. Graham, Review of "Prime Obsession" by J. Derbyshire
S. W. Graham, Review of "The Riemann Hypothesis" by K. Sabbagh
J. P. Gram, Note sur les zéros de la fonction zeta(s) de Riemann, Acta Mathematica, 27 (1903), 289-304.
Mats Granvik, How plot the Riemann zeta zero spectrum with the Fourier transform in Mathematica?
Mats Granvik, Illustration of Andre LeClair's approximation of the initial terms
Mats Granvik, Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros.
Mats Granvik, Plots from the Mathematica programs comparing limiting ratios vs Newton-Raphson method for finding Riemann zeta zeros.
A. Granville, Nombres premiers et chaos quantique (Text in French)
J. Hadamard, Sur la distribution des zeros de la fonction zeta(s) et ses consequences arithmetiques (Text in French)
B. Hayes, The Spectrum of Riemannium
D. R. Heath-Brown, Zeros of the Riemann Zeta-Function on the Critical Line
A. Ivic, On some reasons for doubting the Riemann hypothesis, arXiv:math/0311162 [math.NT], 2003.
A. Ivic, On some recent results in the theory of the zeta-function, arXiv:math/0312425 [math.NT], 2003.
A. Ivic & H. J. J. te Riele, On the zeros of the error term for the mean square of | zeta(1/2 + it) |
D. Jao, PlanetMath.Org, Riemann zeta function
C.-X. Jiang, Disproofs Of Riemann's Hypothesis
N. M. Katz & P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. 36 (1999), 1-26.
E. Klarreich, Prime Time
A. F. Lavrik, Riemann hypotheses
A. LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arXiv:1305.2613 [math-ph], 2013.
N. Levinson, At Least One-Third of Zeros of Riemann's Zeta-Function are on sigma=1/2, Proc Natl Acad Sci U S A. 1974 Apr; 71(4): 1013-1015.
P. Li, On Montgomery's Pair Correlation Conjecture to the Zeros of the Riemann Zeta Function
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R. Meyer, A Spectral Interpretation for The Zeros of The Riemann Zeta Function
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Index entries for zeta function

Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation).

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Table[Round[Im[ZetaZero[n]]], {n, 59}] (* Jean-François Alcover, May 02 2011 *)

apply(round,lfunzeros(lzeta,100)) \\ Charles R Greathouse IV, Mar 10 2016

def A002410_list(n):
    Z = lcalc.zeros(n)
    return [round(z) for z in Z]
A002410_list(59) # Peter Luschny, May 02 2014

a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Charles R Greathouse IV, Sep 14 2012, corrected by Hal M. Switkay, Oct 04 2021

a(n) ~ 2*Pi*(n - 11/8)/ProductLog((n - 11/8)/exp(1)). This is the asymptotic by Guilherme França and André LeClair. - Mats Granvik, Mar 10 2015; corrected May 16 2016

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004

We consider here the imaginary part of 1/2 + iy = z, for which Zeta(z) is a zero.
Note that these are not the first differences of A002410 because rounding is done here AFTER computing the differences. - R. J. Mathar, Jul 04 2009
What is the largest n such that a(n) > 0? - Charles R Greathouse IV, Jan 08 2012
This doesn't seem feasible to compute, probably more than 10^200. - Charles R Greathouse IV, Jan 29 2013

The zeros -2, -4, -6, ... of the Riemann zeta function are considered trivial. The nontrivial zeros are in the "critical strip" 0 < Re(rho_n) < 1. All of the known nontrivial zeros have real part 1/2. In this sequence, we count the prime numbers less than or equal to the imaginary part of these nontrivial zeros.

The Riemann hypothesis (currently unproven) states that all of the nontrivial zeros have real part 1/2.

cp's OEIS Frontend

A002410 Nearest integer to imaginary part of n-th zero of Riemann zeta function.

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A161914 Gaps between the nontrivial zeros of Riemann zeta function, rounded to nearest integers, with a(1)=14.

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A210447 Number of primes <= Im(rho_n), where rho_n is the n-th nontrivial zero of Riemann zeta function.

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A221974 Numbers n such that A161914(n) = 0.

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